normal random variable is a continuous random variable that allows you to manually specify the expectation and variance

constituents

• \(\mu\) the mean
• \(\sigma\) the variance

requirements

\begin{equation} X \sim \mathcal{N}(\mu, \sigma^{2}) \end{equation}

PDF:

\begin{equation} f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} \end{equation}

additional information

normal maximizes entropy

no other random variable uses as little parameters to convey as much information

Use a series of parametrized differentiable + invertible functions to transform simple distributions to complex ones.

\(\text{NP} \cap \text{coNP}: \forall x \in \qty {0,1}^{*}, \exists\) short, efficiently checkable proof of BOTH \(x\) presence/absence in \(L\)

some examples

• in P: PERFECT-MATCHING
• in P: PRIMES
• we don’t know if this is in \(P\): FACTORING … if it was, much of cryptography will break

The Null Space, also known as the kernel, is the subset of vectors which get mapped to \(0\) by some Linear Map.

constituents

Some linear map \(T \in \mathcal{L}(V,W)\)

requirements

The subset of \(V\) which \(T\) maps to \(0\) is called the “Null Space”:

\begin{equation} null\ T = \{v \in V: Tv = 0\} \end{equation}

additional information

the null space is a subspace of the domain

It should probably not be a surprise, given a Null Space is called a Null Space, that the Null Space is a subspace of the domain.

A number can be any of…

• \(\mathbb{N}\): natural number
• \(\mathbb{Z}\): integer
• \(\mathbb{Q}\): rational number
• \(\mathbb{R}\): real number
• \(\mathbb{P}\): irrational number
• \(\mathbb{C}\): complex number